First Exit Times of Non-linear Dynamical Systems in ...

J Stat Phys DOI 10.1007/s10955-010-0041-6

First Exit Times of Non-linear Dynamical Systems in Rd Perturbed by Multifractal Lévy Noise Peter Imkeller · Ilya Pavlyukevich · Michael Stauch

Received: 29 June 2010 / Accepted: 30 July 2010 © Springer Science+Business Media, LLC 2010

Abstract In a domain G ⊂ Rd we study a dynamical system which is perturbed in finitely many directions i by one-dimensional Lévy processes with αi -stable components. We investigate the exit behavior of the system from the domain in the small noise limit. Using probabilistic estimates on the Laplace transform of the exit time we show that it is exponentially distributed with a parameter that depends on the smallest αi . Finally we prove that the system exits from the domain in the direction of the process with the smallest αi . Keywords Lévy process · Lévy flight · First exit time · Exit time law · Perturbed dynamical system · Multifractal noise

1 Introduction. Dynamical Systems Perturbed by Noise The study of the stochastic dynamics of systems with small random perturbations has been receiving much attention in recent years. We consider a finite dimensional dynamical system generated by a vector field b : Rd → Rd via the differential equation dXt0 = b(Xt0 )dt. We assume that the system has at least one asymptotically stable point x. If X 0 starts in the domain of attraction of x, it tends to this point: Xt0 → x as t → ∞.

P. Imkeller · M. Stauch Institut für Mathematik, Humboldt–Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany P. Imkeller e-mail: [email protected] M. Stauch e-mail: [email protected] I. Pavlyukevich () Institut für Stochastik, Friedrich–Schiller–Universität Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany e-mail: [email protected]

P. Imkeller et al.

Now perturb this deterministic system by a small random noise, i.e. consider the stochastic differential equation  t b(Xsε )ds + εψt , (1.1) Xtε = x − 0

where ψ is a d-dimensional random process and ε is small. As an important feature of the asymptotic behavior of the resulting stochastic dynamical system, the exit from an open set G contained in a domain of attraction of a stable point may be investigated. For the most extensively studied case of Gaussian perturbations the theory of large deviations is used to show that the mean exit time is exponential in the noise parameter. More precisely, the logarithmic rate of the mean exit time (Kramers’ time) is proportional to ε12 multiplied by the height of the point of minimal quasipotential on the boundary of G , the latter being related to the vector field b, see [10, 17] . A general theory of large deviations for Markov processes can be found in [23], mean exit times of Markov processes with heavy tails has been first studied in [11]. In recent time, small noise dynamics with stable Lévy noises, especially the first exit problem, has attracted interest in the context modelling of extreme events in climate [8, 12, 18], physics [4] and finance [22]. In case d = 1, and if Gaussian noise is replaced by Lévy noise with discontinuous trajectories, the asymptotic exit characteristics have been studied in [13–15] and in [5–7, 9, 24]. As expected, the presence of jumps allows the process a much faster exit from the domain. In the case of regularly varying tails for the jump measure, the mean exit time depends polynomially on the noise parameter. In this paper we intend to transfer the methods developed in [14] to systems in Euclidean space with finite dimension d. We consider a domain G ⊂ Rd and a dynamical system perturbed by multi-fractal Lévy processes acting in finitely many spatial directions. The result we derive retains the main feature of the one dimensional case. In the small noise limit, we show that the exit time is proportional to the time of the first jump exceeding the distance from the stable point to the boundary of the domain in the direction of the noise component corresponding to the smallest stability index. The structure of the paper is as follows. In Sect. 2 we discuss the geometric framework for the domain G and the dynamical system, and fix the notation. In Sect. 3 we give a heuristic discussion of the asymptotic exit time in terms of its Laplace transform, based on a decomposition of the Lévy noise into a small and a large jump part, which is done in each of the processes’ directions. Section 4 is devoted to a rigorous underpinning of the heuristic frame. We show that small jumps play a marginal role, and establish that jumps in directions of small stability indices happen much earlier than in the remaining ones. In a second part of this section we draw some corollaries concerning the places where exits occur. In the final Sect. 5 we illustrate our findings by some simple examples.

2 Preliminaries and Notation 2.1 The Dynamical System and the Domain G We consider a filtered probability space (Ω, F , (Ft )t≥0 , P). We assume that the filtration (Ft )t≥0 fulfills the usual conditions, i.e. it is right-continuous and consists of σ -algebras which are complete with respect to P, see [19]. Let G ⊆ Rd be a bounded domain with

First Exit Times of Non-linear Dynamical Systems in Rd

0 ∈ G . We are given the following family of stochastic differential equations: 

t

Xtε (x) = x +

b(Xsε (x))ds +

N+M 

0

ελi ri Lit ,

ε > 0, x ∈ G , t ≥ 0,

(2.1)

i=1

where N, M ∈ N. Later we will specify some geometrical assumptions on G . The principal goal of our work consists in describing the small noise dynamical behavior of X ε , i.e. its dynamical behavior as ε tends to zero. We are interested in the exit time of X ε from the domain G and the directions of the exit, both as functions of the small noise parameter ε. We compare our stochastic equation (2.1) with the corresponding deterministic equation given by  t b(zs )ds, t ≥ 0. (2.2) zt (x) = x + 0

Let us first discuss the noise term of (2.1). For 1 ≤ i ≤ N + M the ri are different unit vectors, i.e. ri ∈ Rd , ri = 1 and | ri , rj | = 1 if i = j . The λi > 0 are just pre-factors. For αi ∈ (0, 2), 1 ≤ i ≤ N + M, satisfying α1 = α2 = · · · = αN < αN+1 ≤ αN+2 ≤ · · · ≤ αN+M ,

(2.3)

the stochastic processes Li = (Lit )t≥0 are independent 1-dimensional mean-zero Lévyprocesses with symmetric, αi -stable components. Lévy processes are, in general, random processes with independent and stationary increments that are continuous in probability and have right-continuous paths with left limits. They are characterized by their infinitely divisible one-dimensional distributions, with the following Lévy-Khintchin representations Ee

j

iuLt



u2 = exp −t d + t 2

 R\{0}

  iuy e − 1 − iuyI{|y| 1 we state the following assumptions. (A1) For all 1 ≤ i ≤ N + M the set G ∩ Ωi is connected. Therefore we can find numbers ai , bi > 0 and a closed interval Ii := [−ai , bi ], such that for t ∈ (−ai , bi ) we have: t · ri ∈ G . Since 0 ∈ G and G is open, G ∩ Ωi = ∅. (A2) The vector field b is smooth, i.e. b ∈ C 1 (G ). As a consequence b is Lipschitz continuous: for x, y ∈ G b(x) − b(y) ≤ C x − y .

(2.5)

P. Imkeller et al.

(A3) The boundary ∂ G is a C 1 -manifold, so that the vector field n of the outer normals on the boundary exists. We assume

b(z), n(z) < −

1 , C

(2.6)

for z ∈ ∂ G . This means that b “points into G ”. (A4) For the linearization b(z) = Bz + r(z), z ∈ G , where B is the Jacobian matrix at zero, = 0, r(0) = 0, and that the real parts of the eigenvalues we assume that lim x →0 r(x) x of B are negative and bounded above by − C1 . (A5) Zero is an attractor of the domain, i.e. b(0) = 0, and for every starting value x ∈ G , the deterministic solution (zt (x))t≥0 vanishes asymptotically: lim zt (x) = 0.

t→∞

(2.7)

Since the Lévy processes Lj are semi-martingales, (2.1) is well defined and with the help of (A2) we know that there is a unique solution which possesses the strong Markov property (see [19]). For technical purposes we draw some easy conclusions from these assumptions. We define the inner parts of G by Gδ := {z ∈ G : dist(z, ∂ G ) ≥ δ}. We can find a δ0 > 0, such that if x < δ0 , then x ∈ G and for all δ ∈ (0, δ0 ) and for the constant C from (A1)– (A4) the following is true. (C1) From the theory of ODE and (A4) we know that 0 is an exponentially stable point, i.e. 1 for t ≥ 0 and x ≤ δ0 we have zt (x) < Ce− C t x . (C2) For y < δ0 we consider the straight lines in the direction ri starting from y: gyi (t) = y + t · ri for t ∈ R. We denote the distance function to the boundary by:

(C3)

(C4) (C5) (C6)

di+ (y) := inf{t > 0 : gyi (t) ∈ ∂ G },

(2.8)

di− (y) := sup{t < 0 : gyi (t) ∈ ∂ G }.

(2.9)

With the help of (A1) and the relative compactness of G we infer that gyi (t) ∈ / G for t ∈ / (di− (y), di+ (y)) for all i = 1, . . . , N + M. Otherwise gyi (t) ∈ G . By (A3) the distance functions are continuous. Therefore for all x , y < δ0 we have |di+ (x) − di+ (y)| ≤ C x − y , |di− (x) − di− (y)| ≤ C x − y . For abbreviation we set di+ := di+ (0) and di− = di− (0). For δ ∈ (0, δ0 ) we define δ-tubes outside of G by Ωi+ (δ) := {y ∈ Rd : y, ri ri − y < δ, y, ri  > 0} ∩ G c . Analogously we define Ωi− (δ) by changing to y, ri  < 0. Then we have Ωi− (δ) ∩ Ωj− (δ) ∩ Gδc = ∅ and Ωi+ (δ) ∩ Ωj+ (δ) ∩ Gδc = ∅ for i = j , i, j ∈ {1, . . . , N + M}. [1] states that the sets Gδ are positively invariant for all δ ∈ (0, δ0 ), in the sense that the deterministic solutions starting in Gδ do not leave this set for all times t ≥ 0. As a consequence of (C1), there exists a time T0 > 0 with supx∈G inf{t > 0 : zt (x) ≤ δ0 } ≤ T0 . From (A3), we have supx∈G \Gδ inf{t > 0 : zt (x) ∈ Gδ } ≤ Cδ, for all δ ∈ (0, δ0 ), i.e. the time it takes z, started in G \Gδ , to attain Gδ , is of order δ.

2.2 Decomposition into Small and Large Jump Parts The aim of this subsection is to decompose the Lévy processes Li by means of the LévyItô-decomposition into a small jump part and an independent part with large jumps. While

First Exit Times of Non-linear Dynamical Systems in Rd

small jumps do not contribute much, we will show that large jumps play the crucial role in the exit behavior of X ε in the small noise limit. For ε < 1 and ρ ∈ (0, 1) we consider the truncated Lévy measures    1 1 νξ i (A) := νi A ∩ − ρ , , λi ε λi ε ρ (2.10)    1 c 1 , νηi (A) := νi A ∩ − ρ , λi ε λi ε ρ where i ∈ {1, . . . , N + M} and A ∈ B(R). Thus, by writing the equation Li = ξ i + ηi we have decomposed Li into two processes ξ i and ηi with generating triplets (d, νξi , 0) and (0, νηi , 0) respectively. The absolute values of jumps of the processes εξ i do not exceed the threshold ε 1−ρ . For the finite Lévy measure νηi and for i = 1, . . . , N + M we define the jump intensity  2 α βi := νηi (R) = νi (dx) = λi i ε ραi . (2.11) αi R\[− λ 1ερ , λ 1ερ ] i

i

ηi is a compound Poisson process with intensity βi , and its jumps are distributed according to the law βi−1 νηi (·). For k ≥ 0 denote by τki the kth jumping time of ηi and by Wki := ητi i − ητi i − the heights of the jumps of ηi at time τki . Then the inter-jump k

k

i periods Ski := τki − τk−1 are exponentially distributed with parameter βi . The families i i i ((Wk )k∈N , (Sk )k∈N , (ξt )t≥0 )1≤i≤M+N are independent. For a, b > 0 and min(a, b) > ε 1−ρ we compute the probability for a jump leaving the interval [−a, b] by:  1 dy / [−a, b]) = I{|y|>λ−1 ε−ρ } (y) 1+α P(ελi W1i ∈ i −a b βi R\[ ελ , ελ ] |y| i i i  ε αi αi 1 1 = λi + . (2.12) αi βi a αi bαi

To abbreviate relevant parameters of the compound Poisson part of our dynamics we set i := β := S

1 1 + + α , (−di− )αi (di ) i M+N 

mi := mi (ε) :=

N 

ε αi αi λ i , αi i

N ε α1  α1 mε := mi = λ i , α1 i=1 i i=1

βi ,

i=1

(2.13)

where the di± are chosen according to (C2). Then we see that P(ελi W1i ∈ / [di− , di+ ]) =

mi . βi

(2.14)

Now we define the time of big jumps of the process X ε recursively for k > 1 starting with τ0∗ := 0: τ1∗ :=

N+M

i=1

τ1i ,

τk∗ :=

∗ τji >τk−1

τji .

(2.15)

P. Imkeller et al.

Because of the independence of {τji : i ∈ {1, . . . , N + M}, j ∈ N} we know that τ1∗ has an exponential distribution with parameter β S . Due to the strong Markov property of X ε the ∗ inter-jump times Sk∗ := τk∗ − τk−1 , k ≥ 1, have the same distribution as τ1∗ . Finally we define ε σε the exit time of X from the domain G by / G }. σε = inf{t ≥ 0 : Xtε ∈

(2.16)

As a crucial feature of our reasoning, we have a separate view on the different directions ri . If we compare the mean waiting times for the next jump βi−1 for small ε, we see α

βi−1 < βj−1

⇔

αi λj j α

αj λi i

< ε ρ(αi −αj )

⇔

αi < αj ,

(2.17)

for i = j. This means that for small ε, directions i with smaller αi jump earlier on the average. The same is true if we compare the intensity of big jumps (2.14). Again directions i with smaller αi contribute a larger jump intensity. So we end up with the conclusion that in the limit ε → 0 the process X ε exits the domain in the direction i corresponding to the smallest αi . 3 Heuristic Derivation of the Main Result In this section we give a heuristic sketch of the derivation of our main results on the asymptotics of the exit time σε and the place, where the exit occurs. For this purpose we calculate the Laplace transform of the normalized exit time mε σε . We aim at showing that it belongs to a standard exponentially distributed random variable. This will justify that we can approximate the law of σε by an exponential distribution with intensity mε . We will show below that exits from the domain at times that do not coincide with large jumps are asymptotically negligible. If we take into account exits only at times of large jumps, we can employ the strong Markov property to argue for starting state x ∈ G and θ > −1 in the following way: E[e−θmε σε ] ≈

∞  k=1

≈

∞ 

E[e−θmε σε I{σε = τk∗ }]  ∗

E e−θmε τk

k−1

I{Xt+τj −1 (Xτj −1 ) ∈ G , t ∈ [0, Sj∗ ]}

j =1

k=1

× I{Xt+τk−1 (Xτk−1 ) ∈ G , t ∈ ≈

[0, Sk∗ ), Xτk∗

∈ / G}

∞  ∗ (E[e−θmε τ1 (1 − I{Xτ1∗ ∈ / G })])k−1 k=1 ∗

× E[e−θmε τ1 I{Xτ1∗ ∈ / G }].

(3.1)

To compute the factors in the last expression we use a decomposition into the different directions: ∗

E[e−θmε τ1 I{Xτ1∗ ∈ / G }] =

M+N  l=1

l

E[e−θmε τ1 I{Xτ1∗ ∈ / G }I{τ1l = τ1∗ }]

First Exit Times of Non-linear Dynamical Systems in Rd

≈

M+N 

l

E[e−θmε τ1 I{τ1l = τ1∗ }]

l=1

× P(ελl W1l ∈ / [dl− , dl+ ]) =

M+N  l=1

mε βl ml = S (1 + Rε ). β S + θ mε β l β + θ mε

(3.2)

 ε αi αi The term Rε := m1ε M i=N+1 αi λi i converges to zero as ε → 0, since α1 < αi for i = N + 1, . . . , M. So we can conclude for the Laplace transform in the small noise limit: ∞  S  β − mε (1 + Rε ) k−1

E[e−θmε σε ] ≈

βS

k=1

+ θ mε

βS

mε (1 + Rε ) + θ mε

1 + Rε ε↓0 1 . → θ + 1 + Rε 1+θ

=

(3.3)

With this knowledge we can derive a result on the place where the process exits from the domain. Using (C3), for δ ∈ (0, δ0 ) and i ∈ {1, . . . , M + N } we first obtain

P(Xτε ∗ ∈ Ωi+ (δ)) =

N+M 

1

P(Xτε k ∈ Ωi+ (δ))P(τ1k = τ1∗ ) 1

k=1

= P(Xτε i ∈ Ωi+ (δ))P(τ1i = τ1∗ ) ≈ 1

βi P(ελi W1i ≥ di+ ) βS

1 αi αi + −αi ε λi (di ) . β S αi

=

(3.4)

Employing the calculation that led to the Laplace transform above for θ = 0 we deduce P(Xσε (ε) ∈ Ωi+ (δ)) ≈

∞  ∗ (E[e−θmε τ1 (1 − I{Xτ1∗ ∈ / G })])k−1 P(Xτε ∗ ∈ Ωi+ (δ)) 1

k=1

=

∞  

1−

k=1

=

ε αi αi

k−1 1 αi αi + −αi mε (1 + R ) ε λi (di ) ε S S β β αi

λi i (di+ )−αi α

mε (1 + Rε )

(3.5)

.

Since for small ε mε is of order ε α1 we conclude

P(Xσε (ε)

∈

Ωi+ (δ))

≈

ε αi αi

λi i (di+ )−αi



α

mε (1 + Rε )

ε↓0

→

λ 1 (d + )−α1 i N i α1 , k=1 λk k

i ∈ {1, . . . , N },

0,

else.

α

(3.6)

P. Imkeller et al.

Fig. 1 Simulated trajectories of the jump diffusion X ε driven by two 1.25-stable processes L1 and L2 (l.) and by a 1.25-stable processes L1 and a 1.75-stable process L2 (r.) in directions r1 = ( √1 , √1 ) and 2

2

r2 = (− √1 , √1 ) respectively. In the secons case, the driver L1 with the smaller stability index determines 2

2

the exit

For i ∈ {1, . . . , N } we define ps :=

N 

λk 1 ((di+ )−α1 + (−di− )−α1 ), α

k=1

pi+

(3.7)

:= λi 1 · (di+ )−α1 , α

pi− := λi 1 · (−di− )−α1 , α

and conclude that for every δ ∈ (0, δ0 ) and initial state x ∈ G lim P(Xσε (ε) ∈ Ωi+ (δ)) =

ε→0

lim P(Xσε (ε) ε→0

∈

Ωi− (δ))

pi+ , ps

p− = i , ps

(3.8)

for i = 1, . . . , N and N  p+ + p− i

i=1

i

ps

= 1.

(3.9)

Therefore, X ε exits from the domain G in a little tube in the directions of the smallest α (see Fig. 1). In the remaining sections of this paper we will make the above heuristic arguments rigorous. We proceed in four steps. First step: We prove that as a consequence of exponential stability (C1) the waiting time for a large jump is longer than the time for X ε to enter a small δ0 -ball around zero. If the starting point x has a certain distance to the boundary of G and keeps this distance such that small jumps can be ignored, X ε will leave the domain with a large jump near zero (compare (C4)).

First Exit Times of Non-linear Dynamical Systems in Rd

Second step: Due to the strong Markov property of X ε , we will see that it is enough to do step one at time τ1∗ . Third step: Since the distance functions to the boundary of G are smooth enough we can estimate a jump from a place close to zero by a jump from zero. So the probability of exiting in direction i will be seen to be proportional to νi (R\[di− , di+ ]), i.e. to correspond to a large jump from zero out of the domain. Fourth step: For small ε the processes with αi -stable jump component in direction i with the smallest αi are dominating. Therefore, as we will see, exit time and exit place are determined by the sum of the νi (R\[di− , di+ ]) of the first N directions (compare (C3)). 4 The Main Result: Exit from the Domain G Let us now complement the heuristic arguments of the preceding section with rigorous proofs. We state the main theorems of this paper. For ε > 0, γ > 0 we define the following interior parts Gk of the domain G : Gk := Gk (γ , ε) = {x ∈ G : dist(x, ∂ G ) ≥ (k + C)ε γ },

k = 1, 2, 3, 4.

(4.1)

We consider the exit time σε from the set G1 , defined by σε := σε (γ , x0 ) = inf{t ≥ 0 : Xtε (x0 ) ∈ / G1 },

(4.2)

for x0 ∈ G1 . For 0 < δ < δ0 and i ∈ {1, . . . , M + N } we also define the small tubes in direction ±ri Ωiε+ (δ) := {y ∈ Rd : ∃z ∈ K(δ), t > di+ (z) − (1 + Cr )ε γ : y = t · ri + z}, Ωiε− (δ) := {y ∈ Rd : ∃z ∈ K(δ), t < di− (z) + (1 + Cr )ε γ : y = t · ri + z}.

(4.3)

In these terms we prove the following theorem. Theorem 1 There exists a constant γ0 > 0, such that for γ ∈ (0, γ0 ), θ > −1 and starting points x0 ∈ G the following holds true: 1 θ +1

(4.4)

lim P(mε σε > t) → e−t

(4.5)

lim Ee−θmε σε = ε↓0

and thus for any t ≥ 0 ε↓0

with mε defined in (2.13), that is in the small noise limit the normalized exit time mε σε is distributed with mean 1. Moreover, for any p > 0 we have E(mε σε )p →  ∞exponentially p −y dy. Furthermore, for every δ ∈ (0, δ0 ) the probabilities to exit in direction ±ri 0 y e are given by lim P(Xσεε ∈ Ωiε+ (δ)) = ε↓0

lim P(Xσεε ε↓0

∈

Ωiε− (δ))

pi+ , ps

p− = i , ps

(4.6)

P. Imkeller et al.

for x ∈ G2 and 1 ≤ i ≤ N . In particular we have N  1 + (pi + pi− ) = 1. p s i=1

(4.7)

For technical reasons we formulate the following two propositions for the Laplace transform of the normalized exit time. Theorem 1 is their immediate consequence. Proposition 1 For every θ > −1 and K > 0 there exists ε0 > 0, such that for every ε ∈ (0, ε0 ]: Ee−θmε σε ≥

1−K , 1+θ +K

(4.8)

for starting points x ∈ G2 . Proposition 2 For every θ > −1 and K ∈ (0, θ + 1) there exists ε0 > 0, such that for every ε ∈ (0, ε0 ] Ee−θmε σε ≤

1+K , 1+θ −K

(4.9)

for starting points x ∈ G2 . In the proofs, it will be convenient to decompose the process X ε into a part driven by the small jump component alone, and a complementary large jump part. For this purpose, for t ≥ 0, ε > 0 and x0 ∈ G we define the small jump part by lt :=

N+M 

λi ri ξti ,

(4.10)

i=1

and the corresponding small jump component of X ε by  t yt (x0 ) := x0 + b(ys )ds + εlt ,

t ≥ 0.

(4.11)

0

Recall the constant C, δ0 and T0 from the assumptions. Fixing q := 2 + C 2 and γ ∈ (0, q1 ), 1 1 for n = min( 3C , 2 ) we choose ε small enough, such that n2 ε γ < δ0 . In these terms we define the relaxation time   n (4.12) TR := sup inf t > 0 : zt (x) ≤ ε γ + T0 . 2 x ≤δ0 Furthermore we consider the following processes, which are defined for t ∈ [0, Sk∗ ], k ∈ N, and i = 1, . . . , N + M: i − ξτi ∗ . ξti,k := ξt+τ ∗ k−1

(4.13)

k−1

ξti,k is a Lévy process and due to the strong Markov property we know that ξ i,k has the same law as ξ i . Additionally we have:  vtk (x) := x +

t

b(vsk )ds + 0

N+M  i=1

ελi ri ξti,k

for t ∈ [0, Sk∗ ],

(4.14)

First Exit Times of Non-linear Dynamical Systems in Rd

ltk :=

N+M 

λi ri ξti,k ,

(4.15)

i=1

for x ∈ G and k ∈ N. We have vt1 (x) = x + the jumps at times τk∗ are described by W1∗ =

N+M 

t 0

b(vs1 )ds + εlt = yt (x) for t ∈ [0, τ1∗ ]. For k ≥ 1

ri λi W1i I{τ1∗ = τ1i },

(4.16)

i=1

Wk∗ =

N+M k 

ri λi Wji I{τk∗ = τji }.

(4.17)

i=1 j =1

The set {(Wk∗ )k∈N , (Sk∗ )k∈N } is independent. In this notation, for x ∈ G the process X ε can be further specified by Xtε (x) = vt1 (x) + εW1∗ I{t = τ1∗ },

t ∈ [0, S1∗ ],

ε 2 1 ∗ ∗ ∗ Xt+τ ∗ (x) = vt (vτ ∗ (x) + εW1 ) + εW2 I{t = S2 }, 1

1

t ∈ [0, S2∗ ],

ε k k−1 ∗ Xt+τ + εWk−1 ) + εWk∗ I{t = Sk∗ }, ∗ (x) = vt (vτ ∗ k−1

(4.18)

t ∈ [0, Sk∗ ].

k−1

For the processes v k , k ∈ N, and y ∈ G we define Ak (y) := {vtk (y) ∈ G1 , t ∈ [0, Sk∗ ), vτk∗ (y) + εWk∗ ∈ G1 }, k

B (y) := k

{vtk (y)

∈ G1 , t ∈

[0, Sk∗ ), vτk∗ (y) + εWk∗ k

∈ / G1 },

k ∗ k ∗ A− k (y) := {vt (y) ∈ G1 , t ∈ [0, Sk ), vτ ∗ (y) + εWk ∈ G2 },

(4.19)

k

A˜ k (y) := {vtk (y) ∈ G1 , t ∈ [0, Sk∗ ), vτk∗ (y) + εWk∗ ∈ G1 \G2 }. k

The analogous sets in the directions i ∈ {1, . . . , M + N } are correspondingly given by A1,i (y) := {vt1 (y) ∈ G1 , t ∈ [0, τ1i ), vτ1i (y) + ελi ri W1i ∈ G1 }, 1

B 1,i (y) := {vt1 (y) ∈ G1 , t ∈ [0, τ1i ), vτ1i (y) + ελi ri W1i ∈ / G1 }, 1

A− 1,i (y)

:=

{vt1 (y)

∈ G1 , t ∈

[0, S1i ), vτ1i (y) + ελi ri W1i 1

∈ G2 },

(4.20)

A˜ 1,i (y) := {vt1 (y) ∈ G1 , t ∈ [0, S1i ), vτ1i (y) + ελi ri W1i ∈ G1 \G2 }. 1

5 Some Preliminary Estimates We fix q := 2 + C 2 and γ ∈ (0, q1 ) with C from the assumptions. Let 



ET (, γ ) = ω ∈ Ω : sup εlt (ω) ≤ ε qγ . t∈[0,T ]

(5.1)

P. Imkeller et al.

Lemma 1 Let T > 0 and γ ∈ (0, q1 ). There exists ε0 > 0, such that for all ε ∈ (0, ε0 ) we have 1 sup yt − zt ≤ ε γ , 2 t∈[0,TR ]

(5.2)

on ET (, γ ), for TR < T . Proof Let x0 ∈ G , ω ∈ ET (, γ ), (yt )t≥0 = (yt (x0 )(ω))t≥0 and l = l(ω). Since b is Lipschitz continuous, we have for t ≥ 0 

t

yt − zt ≤ C

ys − zs ds + εlt .

(5.3)

0

With the help of Gronwall’s lemma we conclude yt − zt ≤ eCt εlt ,

(5.4)

and therefore on ET (, γ ) and by TR ≤ T : sup yt − zt ≤ eC·TR sup εlt ≤ eC(Cγ | ln ε|+T0 ) ε qγ ,

t∈[0,TR ]

t∈[0,TR ]

(5.5)

since TR ≤ Cγ | ln ε| + T0 because of the exponential stability. With the choice of q, there exists an ε0 such that for all ε ∈ (0, ε0 ): sup yt − zt ≤

t∈[0,TR ]

εγ , 2

(5.6)

for TR ≤ T and on ET (, γ ).



Let us next define

 ytk (x) = x +

t

b(ysk )ds + ε(lt − lk ),

(5.7)

k

for k ∈ R, k ≤ t . Then we have ytk (yk (x)) = yt (x) and thus ytk (x) − zt−k (x)

 t    k  =  (b(ys ) − b(zs ))ds + ε(lt − lk ) .

(5.8)

k

By an analogous estimation as in the proof of the previous lemma we find sup t∈[k,k+m]

ytk (x) − zt−k (x) ≤

εγ , 2

(5.9)

on {supt∈[k,k+m] ε(lt − lk ) ≤ ε qγ }, for m > 0. Let γ > 0 and ε0 > 0 be chosen so that this holds true. Lemma 2 Let Tε be a non-negative random variable. For all k ∈ N, r > 0, there exists ε0 > 0, such that for all ε ∈ (0, ε0 )

First Exit Times of Non-linear Dynamical Systems in Rd



εγ 2

sup yt (x) − zt (x) ≥

t∈[0,Tε ]



k ⊆ Tε ≥ r ε

 ∪

k−1 



sup

j j +1 j =0 t∈[ εr , εr ]

 ε(lt − l jr ) ≥ ε qγ .

(5.10)

ε

Proof Let x ∈ G1 and r > 0. We choose ε0 > 0 such that for 0 < ε < ε0 we have TR < ε1r . Then we may decompose   εγ sup yt (x) − zt (x) ≥ 2 t∈[0,Tε ]     k εγ ⊆ Tε ≥ r ∪ sup yt (x) − zt (x) ≥ . (5.11) ε 2 t∈[0, kr ] ε

With the definition of TR we have z 1r (x) ≤ ε obtain y 1r (x) ≤ ε γ and therefore,

εγ 2

. On {supt∈[0, 1r ] yt (x) − zt (x) ≤ ε

εγ 2

} we

ε

 sup yt (x) − zt (x) ≥ t∈[0, εkr ]

⊆

 l−1  k−1   l=0 j =1

εγ 2



εγ sup yt (x) − zt− j −1 (y j −1 (x)) < εr εr 2 t∈[ j −1 , j ] εr

εr



εγ ∩ sup yt (x) − zt− lr (y lr (x)) ≥ ε ε 2 t∈[ lr , l+1 r ] ε

⊆

 l−1 k−1  



ε

{ y jr (x) ≤ ε γ } ε

l=0 j =1



εγ ∩ sup yt (y lr (x)) − zt− lr (y lr (x)) ≥ ε ε ε 2 t∈[ lr , l+1 r ] l εr

ε





ε

εγ ⊆ sup yt (x) − zt (x) ≥ 2 t∈[0, 1r ]





ε

∪

k−1 

l r

sup

l l+1 l=1 t∈[ εr , εr ]

⊆

k−1 

sup

j j +1 j =0 t∈[ εr , εr ]

This completes the proof.

ytε (a) − zt− lr (a) ≥ ε

 ε(lt − l jr ) ≥ ε γ q .

εγ , for some a ∈ R, a ≤ ε γ 2



(5.12)

ε



). Let ε → Tε be a positive function with hε := Lemma 3 Let ρ ∈ (0, 1) and γ ∈ (0, 1−ρ q Tε−1 ε qγ +ρ−1 → ∞ with ε → 0. There exists p0 > 0 and ε0 > 0 such that for all 0 < ε < ε0

P. Imkeller et al.

and 0 < p < p0 :

  P sup εlt ≥ ε qγ ≤ exp(−ε −p ).

(5.13)

t∈[0,Tε ]

Proof Since the εξ i are symmetrical mean zero martingales with compact support, we can use Doob’s inequality for exponential functions of martingales to derive for ε > 0, uε > 0 N+M   i qγ P sup εlt ≥ ε qγ ≤ 2 · e−uε ε sup Eeuε ελi ξt . t∈[0,Tε ]

i=1 t∈[0,Tε ]

(5.14)

The last part of this expression is known from the Lévy-Khintchine representation. Hence we have the following chain of inequalities: i

sup E[euε ελi ξt ]

t∈[0,Tε ]

   b (euε ελi y − 1 − uε ελi yI{|y| < 1})ν(dy) = sup exp (uε ελi )2 t + t 2 t∈[0,Tε ] |y|≤ λ 1ερ i    b ≤ sup exp (uε ελi )2 t + t (1 ∧ y 2 ) exp(uε ε 1−ρ )ν(dy) 2 t∈[0,Tε ] |y|≤ λ 1ερ i    b 2 1−ρ ≤ exp Tε (uε ελi ) + m exp(uε ε ) , (5.15) 2

 where we set m = R (1 ∧ y 2 )ν(dy) < ∞. With h(ε) := N+M 2 λi we see n := i=1 N+M

ε qγ +ρ−1 , Tε

u(ε) := ε ρ−1 ln h(ε) and



sup

i=1 t∈[0,Tε ]

E[exp(uε ελi ξti )]

 b ≤ exp Tε ε (ln h(ε)) n + m(N + M)Tε h(ε) 2 2ρ

2

≤ exp(2Tε mh(ε)(N + M)) ≤ exp(2ε ρ+qγ −1 (N + M)m),

(5.16)

and 0 < p < p0 we have for ε small enough that n b2 (ln h(ε))2 ≤ mh(ε). For p0 := − qγ +ρ−1 2   (5.17) P sup εlt ≥ ε qγ ≤ exp(−ε −p ) t∈[0,Tε ]



for ε small enough. This finishes the proof. For x ∈ G , n ≥ 1 let now   εγ Exn := ω ∈ Ω : sup ytn (x)(ω) − zt (x)(ω) ≤ , 2 t∈[0,Sn∗ ] E :=

(5.18)

Ex1 .

With Lemma 2 we see:  (Exn )c

⊆

Sn∗

k ≥ q ε

 ∪

k−1 

sup

j j +1 j =0 t∈[ εr , εr ]

 ε(ltn − l nj ) ≥ ε qγ . εr

(5.19)

First Exit Times of Non-linear Dynamical Systems in Rd r/2

For 0 < r < min(1 − ρ − qγ , α1 ρ) and kε = [ εβ S ], where [·] denotes the integer part, we can prove the following Lemma. Lemma 4 For every θ > −1 there exists a p > 0 and an ε0 > 0, such that for every ε < ε0 : ∗

E[e−θmε τ1 I{E c }] ≤

βS

βS −p e−ε . + θ mε

(5.20)

Proof Since r < 1 − ρ − qγ , with the help of Lemma 3, the Markov property of the ξ i and the independence of τ1∗ and ξ i , we see that there exists ε1 > 0 and p  > 0, such that for 0 < ε < ε1 , 0 ≤ j ≤ kε − 1:    βS −p  −θmε τ1∗ 1 1 qγ E e I sup ε(lt − l j ) ≥ ε e−ε . (5.21) ≤ S r β + θ m j j +1 ε ε t∈[ , ] εr

εr

With (5.19) we obtain for an 0 < ε0 ≤ ε1 , such that for 0 < ε ≤ ε0 we have β S + θ mε > 0: ∗

E[e−θmε τ1 I{E c }]    kε −θmε τ1∗ ∗ ≤E e I τ1 ≥ r ε  k ε −1  ∗ E e−θmε τ1 I sup +  ≤ ≤

t∈[ εjr , jε+1 r ]

j =0 ∞

β S e−(θmε +β

kε /ε r

βS

S )t

dt + kε ·

ε(lt1

βS

− l j ) ≥ ε 1

qγ



εr

βS −p  e−ε + θ mε

βS S −r −p  (e−(θmε +β )kε ε + kε e−ε ). + θ mε

(5.22)

Since kε = O(ε r/2−α1 ρ ) and β S kε ε −r = O(ε −r/2 ), there exists a p > 0 such that for all 0 < ε < ε0 the last estimate can be bounded above by βS

βS −p e−ε . + θ mε

(5.23) 

Lemma 5 Let y ∈ G2 and γ ∈ (0, q1 ). There exists an ε0 > 0 s.t. for ε ∈ (0, ε0 ), 1 ≤ i ≤ M + N: − + i γ γ 1. I{A− 1,i (y)} ≥ I{ελi W1 ∈ (di + (2 + 2C)ε , di − (2 + 2C)ε )}, − + i 1,i / (di , di )}, 2. I{B (y)} ≥ I{ελi W1 ∈

on E ∩ {TR < τ1i }. γ

Proof If ε0 < δ0 , we can easily check the estimate taking into account that on the events E  and {TR < τ1i } we have yτ i ≤ ε γ . Remember C > 1. 1

Lemma 6 Let γ ∈ (0, q1 ). For y ∈ G , 1 ≤ i ≤ M + N , there is an ε0 > 0, such that for all ε ∈ (0, ε0 ) i γ 2 γ i A− 1,i (y) ⊇ E ∩ {|ελi W1 | < ε } ∩ {5C ε ≤ τ1 }.

(5.24)

P. Imkeller et al. γ

Proof If we choose ε0 > 0 small enough such that (4 + C)ε0 ≤ δ0 , we can conclude for ε < ε0 with the help of (C6) sup inf{t > 0 : zt (x) ∈ G4 } ≤ 5C 2 ε γ ,

x∈G \G4

(5.25)

remembering C > 1. I.e. starting in G2 we know zt ∈ G4 on τ1i > t ≥ 5C 2 ε γ and therefore on  E: yt ∈ G3 ⊂ G2 . The next lemma is proven with the same arguments as the preceding ones. Lemma 7 Let i ∈ {1, . . . , M + N }, γ ∈ (0, q1 ) and y ∈ G1 . There is ε0 > 0, such that for all ε ∈ (0, ε0 ): 1. I{A1,i (y)} ≤ I{ελi W1i ∈ [di− , di+ ]}, / [di− + (1 + 2C)ε γ , di+ − (1 + 2C)ε γ ]}, 2. I{B 1,i (y)} ≤ I{ελi W1i ∈ i 1,i 3. I{A˜ (y)} ≤ I{ελi W1 ∈ [di− , di− + (2 + 2C)ε γ ]} + I{ελi W1i ∈ [di+ − (2 + 2C)ε γ , di+ ]}, on E ∩ {τ1i ≥ TR }.



). For θ > −1 and m > 0 there exist constants Proposition 3 Let ρ ∈ (0, 1) and γ ∈ (0, 1−ρ q p, C1 , C2 , ε0 > 0, such that for ε ∈ (0, ε0 ): 1.

N+M 

∗

E[e−θmε τ1 I{τ1i = τ1∗ }I{τ1∗ < TR }I{|ελi W1i | ≥ ε γ }]

i=1

≤ C1 ε α1 (1−ρ−γ ) 2. 3.

N+M  i=1 N+M 

E[e

−θmε τ1∗

(β S )2 | ln(ε)| , θ mε + β S

I{τ1i

=

τ1∗ }I{E c }]

∗

βS ≤ S exp(−ε −p ), β + θ mε

E[e−θmε τ1 I{τ1i = τ1∗ }I{τ1i < mε γ }] ≤ C2

i=1

(5.26)

βS β S εγ . β S + θ mε

Proof A straightforward calculation yields that for two independent exponentially distributed r.v. X, Y with parameters α > 0 resp. β > 0 E[e−θX I{X ≤ Y }I{X < k}) =

α (1 − e−(α+β+θ)k ), α+β +θ

(5.27)

for k > 0 and θ > min(−α, −β, −(α + β)). Additionally, exponential stability and the definition of TR imply that there is ε0 > 0 such that for 0 < ε < ε0 and a constant C1 > 0: (β S + θ mε )TR ≤ (β S + θ mε )(C| ln ε| + T0 ) ≤ C1 β S | ln ε|.

(5.28)

We use this together with the independence and the law properties of τ1i and W1i to prove the proposition.  Let Ii := [di− +(2+2C)ε γ , di+ −(2+2C)ε γ ]. The following two propositions are proved as the preceding one, taking into account the properties of τ1i and W1i and the following

First Exit Times of Non-linear Dynamical Systems in Rd

argument. Let ε be small enough such that min(di+ , −di− ) > ε 1−ρ and for a fixed K > 0: Kε γ < min(−di− , di+ ). Then we estimate P[ελi W1i ∈ [di− + ε γ K, di+ − ε γ K]]

(5.29)

1 (λi ε)αi [(−di− − ε γ K)−αi + (di+ − ε γ K)−αi ] βi αi mi ≥1− (1 + κε l ), βi =1−

(5.30) (5.31)

for a κ > 0, l > 0 and i ∈ {1, . . . , M + N }. Proposition 4 Let ρ ∈ (0, 1) and γ ∈ (0, 1−ρ ). For θ > −1 there are constants ε0 , l, C3 , q C4 > 0, such that for all ε ∈ (0, ε0 ]:

1.

N+M 

i

E[e−θmε τ1 I{τ1i = τ1∗ }I{εW1i λi ∈ Ii }]

i=1

≥ 2.

N+M 

  βS mε l (1 + C ε ) , 1 − 3 β S + mε θ βS

E[e

−θmε τ1i

I{τ1i

>

TR }I{τ1i

=

τ1∗ }I{ελi W1i

(5.32) ∈ /

[di− , di+ ]}]

i=1

≥

mε βS (1 − C4 β S | ln ε|). mε θ + β S β S

). For θ > −1 there exits ε0 > 0 and C5 > 0, Proposition 5 Let ρ ∈ (0, 1) and γ ∈ (0, 1−ρ q such that for all ε ∈ (0, ε0 ):

1. 2.

N+M  i=1 N+M 

E[e

−θmε τ1i

I{τ1i

=

τ1∗ }I{εW1i λi

∈

[di− , di+ ]}]

  βS mε ≤ S 1− S , β + mε θ β

i

E[e−θmε τ1 I{τ1i = τ1∗ }

i=1

× I{ελi W1i ∈ / [di− + (1 + 2C)ε γ , di+ − (1 + 2C)ε γ ]}] ≤ 3.

N+M 

mε βS (1 + C5 ), β S + mε θ β S

E[e−θmε τ1 I{τ1i = τ1∗ }I{ελi W1i ∈ [di− , di− + (2 + 2C)ε γ ]} i

i=1

+ I{ελi W1i ∈ [di+ − (2 + 2C)ε γ , di+ ]}] ≤

mε βS C5 . β S + mε θ β S

(5.33)

P. Imkeller et al.

6 Proof of Proposition 1: The Lower Bound For θ > −1 and x ∈ G2 we have the following estimate: E[e−θmε σε ] ≥

∞ 

∗

E[e−θmε τk I{σε = τk∗ }].

(6.1)

k=1

For k ∈ N we further have ∗

∗

E[e−θmε τk I{σε = τk∗ }] = E[e−θmε τk I{Xtε ∈ G1 , t ∈ [0, τk∗ ), Xτε ∗ ∈ / G1 }] k

= E[e

−θmε τk∗

× I{Xtε

I{Xtε

∈ G1 , t ∈

[0, τ1∗ )} · · ·

∗ ∈ G1 , t ∈ [τk−1 , τk∗ )}I{Xτε ∗ ∈ / G1 }]. k

(6.2)

The strong Markov property allows to write ∗

E[e−θmε τk I{σε = τk∗ }]  k−1 ∗ I{Xtε (Xτε ∗ ) ∈ G1 , t ∈ [0, Si∗ ]} = E e−θmε τk i−1

i=1 ∗ ) × I{Xtε (Xτk−1

∈ G1 , t ∈

[0, Sk∗ ), Xτε ∗ k

∈ / G1 } .

(6.3)

The preceding expression can be estimated by applying law properties of the small jump component v k :  ∗

E e−θmε τk

k−1

∗ ∗ ) ∈ G1 , t ∈ [0, S ]} I{Xtε (Xτi−1 i

i=1

∗ ) × I{Xtε (Xτk−1

=E

k−1

∈ G1 , t ∈

[0, Sk∗ ), Xτε ∗ k

∈ / G1 }

e

−θmε Si∗

I{A

i

∗ ∗ )} (Xτε ∗ )}e−θmε Sk I{B k (Xτk−1 i−1

i=1

  k−1   ∗ ∗ ≥ E e−θmε S1 inf I{A− E e−θmε S1 inf I{B 1 (y)} . 1 (y)} y∈G2

y∈G2

(6.4)

The two terms in the last preceding expression will be further estimated by decomposing them into terms reflecting jumps in the various directions. We estimate for y ∈ G2 : I{A− 1 (y)} ≥

N+M 

∗ i I{A− 1,i (y)}I{τ1 = τ1 }.

(6.5)

i=1

Let Ii = (di− + (2 + 2C)ε γ , di+ − (2 + 2C)ε γ ). Using Lemmas 5 and 6 we obtain the following chain of inequalities for A− 1,i , i ∈ {1, . . . , M + N }:

First Exit Times of Non-linear Dynamical Systems in Rd − I{A− 1,i (y)} ≥ I{A1,i (y)}I{E}

≥ I{εW1i λi ∈ Ii } − 2 · I{E c } − I{τ1i ≤ TR }I{|ελi W1i | ≥ ε γ } − I{τ1i < 5C 2 ε γ },

(6.6)

for ε small enough such that [−ε γ , ε γ ] ⊆ Ii . Substituting this into (6.5) gives I{A− 1 (y)} ≥

N+M 

I{τ1∗ = τ1i }(I{εW1i λi ∈ Ii } − I{τ1i < 5C 2 ε γ }

i=1

− 2 · I{E c } − I{τ1i ≤ TR }I{|ελi W1i | ≥ ε γ }).

(6.7)

Hence by Propositions 3 and 4 there exits K > 0 and ε1 > 0, such that for all ε < ε1 :     βS mε ∗ E e−θmε τ1 inf I{A− (y)} ≥ (1 + K) . (6.8) 1 − 1 y∈G2 β S + mε θ βS For the sets B 1 (y), y ∈ G2 , we use similar arguments. We have I{B 1 (y)} =

N+M 

I{B 1,i (y)}I{τ1∗ = τ1i }.

(6.9)

i=1

Lemma 5 allows to write I{B 1,i (y)} ≥ I{B 1,i (y)}I{E}I{τ1i > TR } / [di− , di+ ]}I{τ1i > TR } − I{E c }, ≥ I{εW1i λi ∈ for i ∈ {1, . . . , M + N }. If we insert this to (6.9) we see: I{B 1 (y)} ≥

N+M 

I{τ1∗ = τ1i }(I{εW1i λi ∈ / [di− , di+ ]}I{τ1i > TR } − I{E c }).

(6.10)

i=1

Again with (6.10) and Proposition 4 for K > 0 there exists ε2 > 0, such that for all ε ∈ (0, ε2 ):   mε βS ∗ (1 − K). (6.11) E e−θmε τ1 inf I{B 1 (y)} ≥ S y∈G2 β + mε θ β S Finally for K > 0 we conclude with the help of (6.4), (6.8) and (6.11) that there exists an ε (1 + K) < 1 holds, so that ε0 > 0 with 0 < ε < ε0 ≤ min(ε1 , ε2 ) and small enough, that m βS we have k−1  k  ∞   βS mε mε −θmε σε E[e ]≥ (1 − K) 1 − S (1 + K) β S + mε θ β βS k=1 = This justifies the claim.

1−K . θ +1+K

(6.12) 

P. Imkeller et al.

7 Proof of Proposition 2: The Upper Bound As in the proof of Proposition 1 we decompose the Laplace transform into E[e−θmε σε ] =

∞ 

E[[e−θmε σε I{σε = τk∗ }] +Restk ],    k=1

(7.1)

(∗)

where

 Restk ≤

∗

∗ , τk∗ )}], E[e−θmε τk I{σε ∈ (τk−1

E[e

∗ −θmε τk−1

θ ∈ (−1, 0)

∗ I{σε ∈ (τk−1 , τk∗ )}],

θ ∈ [0, ∞).

(7.2)

Let us decompose (∗) similarly to the proof of Proposition 1. This yields the following estimates: E[e−θmε σε I{σε = τk∗ }] ∗

/ G1 }] = E[e−θmε τk I{Xtε ∈ G1 , t ∈ [0, τk∗ ); Xτε ∗ ∈ k  k−1 ∗ I{Xtε (Xτε ∗ ) ∈ G1 , t ∈ [0, Si∗ )} = E e−θmε τk i−1

i=1

× I{Xtε (Xτε ∗ ) k−1 =E

k−1

∈

G1 ; Xτε ∗ k

∈ / G1 }

e

−θmε Si∗

∗ )}e I{A (Xτi−1

i

−θmε Sk∗

∗ )} I{B (Xτk−1

k

i=1

k−1     ∗ ∗ E e−θmε τ1 sup I{B 1 (y)} . ≤ E e−θmε τ1 sup I{A1 (y)} y∈G1 y∈G1      (O1)

(7.3)

(O2)

For k = 1 and x ∈ G2 we get in the cases θ ∈ (−1, 0) resp. θ > 0  Rest1 =  ≤  ≤

∗

E[e−θmε τ1 I{σε ∈ (0, τ1∗ )}], E[I{σε ∈ (0, τ1∗ )}], ∗

E[e−θmε τ1 I{∃t ∈ (0, S1∗ ) : Xtε ∈ / G1 }], / G1 }], E[I{∃t ∈ (0, S1∗ ) : Xtε ∈ ∗

E[e−θmε τ1 supy∈G2 I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ / G1 }], / G1 }]. E[supy∈G2 I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈

For k ≥ 2 we obtain, in the cases θ ∈ (−1, 0) resp. θ > 0  Restk =

∗

∗ , τk∗ )}], E[e−θmε τk I{σε ∈ (τk−1 ∗

∗ , τk∗ )}], E[e−θmε τk−1 I{σε ∈ (τk−1

(7.4)

First Exit Times of Non-linear Dynamical Systems in Rd

 =

≤

∗

∗ ∗ E[e−θmε τk I{Xtε ∈ G1 , t ∈ [0, τk−1 )}I{∃t ∈ (τk−1 , τk∗ ) : Xtε ∈ / G1 }] ∗

∗ ∗ )}I{∃t ∈ (τk−1 , τk∗ ) : Xtε ∈ / G1 }] E[e−θmε τk−1 I{Xtε ∈ G1 , t ∈ [0, τk−1 ⎧ −θmε S1∗ ⎪ supy∈G1 I{A1 (y)}]]k−2 ⎪ ⎪[E[e ⎪ ⎪ ∗ ⎨E[e−θmε S1 sup I{A1 (y)}I{∃t ∈ (0, S ∗ ) : v 2 (X ε ∗ + εW ∗ ) ∈ / G }] y∈G1

2

t

τ1

1

1

∗ ⎪ ⎪ [E[e−θmε S1 supy∈G1 I{A1 (y)}]]k−2 ⎪ ⎪ ⎪ ε ∗ ∗ 1 2 ⎩E[e−θmε S1∗ sup / G1 }}. y∈G1 I{A (y)}I{∃t ∈ (0, S2 ) : vt (Xτ ∗ + εW1 ) ∈

(7.5)

1

Estimation of the part (O1) For y ∈ G1 we decompose I{A1 (y)} =

N+M 

I{A1,i (y)}I{τ1∗ = τ1i },

(7.6)

i=1

which can be further estimated with Lemma 7 for 1 ≤ i ≤ M + N by I{A1,i (y)} ≤ I{A1,i (y)}I{E} + I{E c } $ % ≤ I ελi W1i ∈ [di− , di+ ] I{E}I{|ελi W1i | > ε γ }I{τ1i ≥ TR } + I{|ελi W1i | > ε γ }I{τ1i < TR } + I{E c } + I{|ελi W1i | ≤ ε γ }.

(7.7)

If we choose ε small enough such that from |ελi W1i | ≤ ε γ we can deduce ελi W1i ∈ [di− , di+ ] we get $ % I{A1,i (y)} ≤ I ελi W1i ∈ [di− , di+ ] + I{E c } + I{|ελi W1i | > ε γ }I{τ1i < TR }.

(7.8)

With the help of Proposition 3 and 5 for every K > 0 there is an ε1 > 0, such that for all ε ∈ (0, ε1 ) the inequality    βS mε ∗ E e−θmε τ1 sup I{A1 (y)} ≤ (1 − K) . (7.9) 1 − θ mε + β S βS y∈G1 holds. Estimation of the part (O2) Again for y ∈ G1 we have the decomposition I{B 1 (y)} =

N+M 

I{B 1,i (y)}I{τ1i = τ1∗ }.

i=1

With the help of Lemma 7 we see I{B 1,i (y)} ≤ I{B 1,i (y)}I{E} + I{E c } $ % ≤ I ελi W1i ∈ / [di− + (1 + 2C)ε γ , di+ − (1 + 2C)ε γ ] + I{|ελi W1i | > ε γ }I{τ1i < TR } + I{ET (ε, γ )c } + I{τ1i ≥ TR } + I{τ1i < 5C 2 ε γ },

(7.10)

P. Imkeller et al.

where in the last step we use that I{B 1,i (y)}I{|ελi W1i | ≤ ε γ }I{τ1i ≥ 5C 2 ε γ } = 0. For ε small enough we deduce for a constant K > 0   ∗ E e−θmε τ1 sup I{B 1 (y)} ≤ y∈G1

 βS mε K 1+ . θ mε + β S β S 3

(7.11)

Estimation of the remainder terms / G1 } for y ∈ G2 , write To estimate I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ / G1 } =

N+M 

(7.12)

I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ / G1 }I{τ1∗ = τ1i }

(7.13)

& ' I{τ1∗ = τ1i } I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ / G1 }I{E} + I{E c }

(7.14)

i=1

≤

N+M  i=1

≤ I{E c }.

(7.15)

For a K > 0 there exits ε3 > 0, such that for all 0 < ε < ε3 & ' & ' ∗ ∗ E e−θmε τ1 I{∃t ∈ (0, S1∗ ) : vt1 (y) ∈ / G1 } ≤ E e−θmε τ1 I{ET (ε, γ )c } βS exp(−ε −p ) θ mε + β S   mε K βS . ≤ θ mε + β S β S 3 ≤

(7.16)

For k = 1 we can conclude  Rest1 ≤

βS mε K ( ), θmε +β S β S 3 −p

exp(−ε

),

θ ∈ (−1, 0) θ ∈ [0, ∞]

≤

  mε K βS . θ mε + β S β S 3

(7.17)

In case k ≥ 2 we may estimate for y ∈ G1 : / G1 } I{A1 (y)}I{∃t ∈ (0, S2∗ ) : vt2 (Xτε ∗ + εW1∗ ) ∈ 1

≤ sup I{∃t ∈ y∈G2

(0, S2∗ )

:

vt2 (y)

∈ / G1 }

$ % + I vt1 (y) ∈ G1 , t ∈ [0, τ1∗ ], vτ1∗ + εW1∗ ∈ G1 \G2 1

≤ sup I{∃t ∈ y∈G2

(0, S1∗ )

:

vt1 (y)

∈ / G1 }

$ % + I vt1 (y) ∈ G1 , t ∈ [0, τ1∗ ], vτ1∗ + εW1∗ ∈ G1 \G2 . 1

(7.18)

The first term is identical to the expression for k = 1. The second part is treated as (O1), where I{A1 (y)} is replaced with I{A˜ 1 (y)}.

First Exit Times of Non-linear Dynamical Systems in Rd

Therefore, in case θ ≥ 0 for K > 0 there exists an ε4 > 0 such that for all 0 < ε < ε4 :   ∗ E e−θγ τ1 sup I{A1 (y)}I{∃t ∈ (0, S2∗ ) : vt2 (Xτε ∗ + εW1∗ ) ∈ / G1 } 1

y∈G1

≤

  βS mε −p α1 (1−ρ−γ ) S ) + C + C ε (β )| ln ε| 2 exp(−ε 5 1 θ mε + β S βS

≤

βS mε K . θ mε + β S β S 3

(7.19)

Analogously, for θ ∈ (−1, 0) and K > 0 there is an ε5 > 0, such that for all 0 < ε < ε5 : ' & ∗ ∗ / G1 } E e−θγ τ1 +τ2 sup I{A1 (y)}I{∃t ∈ (0, S2∗ ) : vt2 (Xτε ∗ + εW1∗ ) ∈ 1

y∈G1

 ≤ ≤

βS θ mε + β S

2

mε K βS 4

βS mε K . θ mε + β S β S 3

(7.20) S

If we choose ε > 0 so that 0 < ε < ε0 ≤ min(ε1 , ε2 , ε3 , ε4 , ε5 ) and that θmβ+β S [1 − ε mε (1 − C)] < 1, we may estimate with (7.9), (7.11), (7.17), (7.19) and (7.20) for K ∈ βS (0, θ + 1) to obtain the desired inequality & ' E e−θmε σε  k−1   ∞   βS mε βS mε K ≤ 1 − S (1 − K) 1+ θ mε + β S β θ mε + β S β S 3 k=1 k−2  ∞  βS mε C  βS mε K βS mε + (1 − K) + 1 − S S S βS 3 θ mε + β S β S 3 θ m + β β θ m + β ε ε k=2 =

1+K . θ +1−K

(7.21)

This completes the proof.  To obtain Theorem 1 we conclude directly from Propositions 1 and 2 that for K ∈ (0, θ + 1): 1+K 1−K ≤ Ee−θmε σε ≤ . θ +1+K θ +1−K

(7.22)

Since K can be chosen arbitrarily, we see lim Ee−θmε σε = ε↓0

1 . θ +1

(7.23)

Convergence of moments and exit probabilities Since the Laplace transform is finite for −1 < θ < 0, the random variable mε σε has moments of all orders. Since the family {(mε σε )p }0 0). (8.1) Xt (x) = x − 0

Here L is a Lévy process with an α-stable component and U ∈ C 1 is a ‘parabolic’-shaped potential function, i.e. U has a global minimum at the origin, U  (x)x ≥ 0, U (0) = 0, U  (x) = 0 iff x = 0 and U  (0) = M > 0. This guarantees the existence of a unique solution, that 0 is an exponentially stable point and an attractor of the domain. So all the required assumptions on the domain and the vector field are fulfilled. Thus, the exit time / [−b, a]} is approximately exponentially distributed with parameter σ (ε) = inf{t > 0 : Xtε ∈ εα 1 [ + b1α ]. For the probability on which side of the interval the process exits, we have α aα lim P(Xσεε > a) =

ε→0

lim P(Xσεε < −b) =

ε→0

a −α , + b−α

a −α

b−α . −α a + b−α

(8.2)

This result is exactly the one obtained in [13]. Ornstein-Uhlenbeck-type processes We consider the ball of radius R in Rd and for θ > 0 the SDE dXtε = −θ Xtε + ε(r1 dL1t + r2 dL2t )

(ε > 0),

(8.3)

where L1 and L2 are Lévy processes with α-stable components. Since all the assumptions are fulfilled, we know that the exit time is approximately exponential distributed with para4ε α 1 meter αR α and the probability of leaving the ball in each of the four directions is 4 . Jumps dominate the Brownian motion This example deals with the intuition that if the system is perturbed by a Lévy noise with stable component and a Brownian motion, in the small limit of ε the Lévy noise will dominate its exit behavior. For this purpose we define the dynamical system dXtε = b(Xtε ) + ε(λ1 r1 dBt + λ2 r2 dLt ),

(8.4)

where L is a Lévy process with α-stable component as before, B an independent standard Brownian motion and b a vector field that fulfills the conditions above. Let us define

First Exit Times of Non-linear Dynamical Systems in Rd

ξ˜t := ε(λ1 r1 Bt + λ2 r2 ξt ), where ξ is the small jump process of L. We can easily repeat the arguments of lemma 3 to show   P sup ε ξ˜t ≥ ε qγ ≤ exp(−ε −p ). (8.5) t∈[0,Tε ]

This means that we can consider the Brownian motion to be part of the small jump part. Only the big jumps of L are important for the exit.

9 Conclusions In this paper we study random perturbations of multidimensional deterministic dynamical systems with a stable attractor at the origin by white Lévy noise of small amplitude. The Lévy noise is assumed to be multifractal, i.e. it is composed of a finite sum of independent symmetric one-dimensional αi -stable processes with laws being supported on straight lines spanned on unit vectors ri . In the small amplitude limit we solve the problem of the first exit of the perturbed system from a bounded domain around the origin. We prove that the first exit time is exponentially distributed, determine the explicit formula for its mean value (Kramers’ time) and in general all higher moments. Furthermore we determine the asymptotic law of the location of the first exit. It turns out that the exit is essentially controlled by the Lévy noise with the smallest stability index αmin and the spatial geometry of the domain, in particular the length of the segments of the straight line spanned on the vector rαmin connecting the stable attractor and the domainn’s border. Our proof is based on a simultaneous decomposition of the driving processes into appropriate small and big jump parts. Acknowledgements The authors acknowledge financial support by DFG within SFB 555 collaborative research project. The authors are also grateful R. Schilling for discussions motivated the subject of this paper and to two anonymous referees for the carefull reading of the manuscript and their valuable comments.

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